Problem: Solve for $x$ and $y$ using elimination. $\begin{align*}8x+3y &= 1 \\ 8x+6y &= -2\end{align*}$
Answer: We can eliminate $x$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $-1$ and the bottom equation by $1$ $\begin{align*}-8x-3y &= -1\\ 8x+6y &= -2\end{align*}$ Add the top and bottom equations. $3y = -3$ Divide both sides by $3$ and reduce as necessary. $y = -1$ Substitute $-1$ for $y$ in the top equation. $8x+3( -1) = 1$ $8x-3 = 1$ $8x = 4$ $x = \dfrac{1}{2}$ The solution is $\enspace x = \dfrac{1}{2}, \enspace y = -1$.